Since we are dealing with bounded solutions,
then this function and all other test functions used in the article
belong to \(W^{1,G}_{0}(\Omega )\) .
This is a consequence of conditions (\({\rm g}_{0}\) ), (\({\rm g}_{1}\) ),
the result of Marcus and Mizel [1]} and the notion of the weak inequality
on the boundary \(\partial \Omega \)  [2]}, [3]}.
So, after substitution (REF ) into (REF ), we have
\(\begin{aligned}&\int _{A_{k,r(1+\sigma )}}\frac{G(x,|\nabla u|)}{\mathcal {G}\left( x_{0}, u_{l,\rho }/\rho \right)}\,\zeta ^{\,q}\,dx\\&+\int _{A_{k,r(1+\sigma )}}\frac{G(x,|\nabla u|)}{\mathcal {G}\left( x_{0}, u_{l,\rho }/\rho \right)}\left\lbrace  \frac{G(x_{0}, u_{l,\rho }/\rho )}{\mathcal {G}(x_{0}, u_{l,\rho }/\rho )}-1 \right\rbrace (w-k)_{+}\,\zeta ^{\,q}\,dx\\&\leqslant \frac{\gamma }{\sigma }\int _{A_{k,r(1+\sigma )}}\frac{g(x,|\nabla u|)}{\mathcal {G}\left( x_{0}, u_{l,\rho }/\rho \right)}\,\frac{u_{l,\rho }}{\rho }\,(w-k)_{+}\,\zeta ^{\,q-1}\,dx,\end{aligned}\) 
